950 Mathematics (M) - Portal Rasmi Majlis Peperiksaan Malaysia

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950 Mathematics (M) - Portal Rasmi Majlis Peperiksaan Malaysia
STPM/S(E)950
MAJLIS PEPERIKSAAN MALAYSIA
(MALAYSIAN EXAMINATIONS COUNCIL)
PEPERIKSAAN
SIJIL TINGGI PERSEKOLAHAN MALAYSIA
(MALAYSIA HIGHER SCHOOL CERTIFICATE EXAMINATION)
MATHEMATICS (M)
Syllabus and Specimen Papers
This syllabus applies for the 2012/2013 session and thereafter until further notice.
NATIONAL EDUCATION PHILOSOPHY
“Education in Malaysia is an on-going effort towards further
developing the potential of individuals in a holistic and
integrated manner, so as to produce individuals who are
intellectually, spiritually, emotionally and physically
balanced and harmonious, based on a belief in and devotion
to God. Such effort is designed to produce Malaysian
citizens who are knowledgeable and competent, who possess
high moral standards, and who are responsible and capable
of achieving a high level of personal well-being as well as
being able to contribute to the betterment of the family, the
society and the nation at large.”
FOREWORD
This revised Mathematics (M) syllabus is designed to replace the existing syllabus which has been in
use since the 2002 STPM examination. This new syllabus will be enforced in 2012 and the first
examination will also be held the same year. The revision of the syllabus takes into account the
changes made by the Malaysian Examinations Council (MEC) to the existing STPM examination.
Through the new system, the form six study will be divided into three terms, and candidates will sit
for an examination at the end of each term. The new syllabus fulfils the requirements of this new
system. The main objective of introducing the new examination system is to enhance the teaching
and learning orientation of form six so as to be in line with the orientation of teaching and learning in
colleges and universities.
The Mathematics (M) syllabus is designed to provide a framework for a pre-university course that
enables candidates to develop the understanding of mathematical concepts and mathematical thinking,
and acquire skills in problem solving and the applications of mathematics related to social sciences
and management. The assessment tools of this syllabus consist of written papers and coursework.
Coursework offers opportunities for candidates to conduct mathematical investigation and
mathematical modelling that enhance their understanding of mathematical processes and applications
and provide a platform for them to develop soft skills.
The syllabus contains topics, teaching periods, learning outcomes, examination format, performance
descriptions and specimen papers.
The design of this syllabus was undertaken by a committee chaired by Professor Dr. Abu Osman bin
Md Tap from International Islamic University Malaysia. Other committee members consist of
university lecturers, representatives from the Curriculum Development Division, Ministry of
Education Malaysia, and experienced teachers who are teaching Mathematics. On behalf of MEC, I
would like to thank the committee for their commitment and invaluable contribution. It is hoped that
this syllabus will be a guide for teachers and candidates in the teaching and learning process.
Chief Executive
Malaysian Examinations Council
CONTENTS
Syllabus 950 Mathematics (M)
Page
Aims
1
Objectives
1
Content
First Term:
Algebra and Calculus
2–4
Second Term: Statistics
5–7
Third Term:
8 – 10
Financial and Decision Mathematics
Coursework
11
Scheme of Assessment
12
Performance Descriptions
13
Mathematical Notation
14 – 17
Electronic Calculators
18
Reference Books
18
Specimen Paper 1
19 – 24
Specimen Paper 2
25 – 38
Specimen Paper 3
39 – 48
Specimen Assignment Paper 4
49 – 50
SYLLABUS
950 MATHEMATICS (M)
[May not be taken with 954 Mathematics (T)]
Aims
The Mathematics (M) syllabus is designed to provide a framework for a pre-university course that
enables candidates to develop the understanding of mathematical concepts and mathematical thinking,
and acquire skills in problem solving and the applications of mathematics related to social sciences
and management.
Objectives
The objectives of the syllabus are to enable candidates to:
(a)
use mathematical concepts, terminology and notation;
(b)
display and interpret mathematical information in tabular, diagrammatic and graphical forms;
(c)
identify mathematical patterns and structures in a variety of situations;
(d)
use appropriate mathematical models in different contexts;
(e)
apply mathematical principles and techniques in solving problems;
(f)
carry out calculations and approximations to an appropriate degree of accuracy;
(g)
interpret the significance and reasonableness of results;
(h)
present mathematical explanations, arguments and conclusions.
1
FIRST TERM: ALGEBRA AND CALCULUS
Topic
1
Teaching
Period
Learning Outcome
Functions
20
Candidates should be able to:
1.1
6
(a) state the domain and range of a function, and
find composite functions;
Functions
(b) determine whether a function is one-to-one,
and find the inverse of a one-to-one function;
(c) sketch the graphs of simple functions,
including piecewise-defined functions;
1.2
Polynomial and
rational functions
8
(d) use the factor theorem and the remainder
theorem;
(e) solve polynomial and rational equations and
inequalities;
(f)
solve equations and inequalities involving
modulus signs in simple cases;
(g) decompose a rational expression into partial
fractions in cases where the denominator has
two distinct linear factors, or a linear factor
and a prime quadratic factor;
1.3
2
Exponential and
logarithmic functions
6
(h) relate exponential and logarithmic functions,
algebraically and graphically;
(i)
use the properties of exponents and logarithms;
(j)
solve equations and inequalities involving
exponential or logarithmic expressions.
Sequences and Series
18
Candidates should be able to:
2.1
4
(a) use an explicit formula and a recursive formula
for a sequence;
Sequences
(b) find the limit of a convergent sequence;
2.2
Series
8
(c) use the formulae for the nth term and for the
sum of the first n terms of an arithmetic series
and of a geometric series;
(d) identify the condition for the convergence of a
geometric series, and use the formula for the
sum of a convergent geometric series;
(e) use the method of differences to find the nth
partial sum of a series, and deduce the sum of
the series in the case when it is convergent;
2
Topic
2.3
Binomial expansions
Teaching
Period
6
Learning Outcome
(f)
expand (a + b)n, where n ∈ ]+ ;
(g) expand (1 + x)n, where n ∈ _, and identify the
condition | x | < 1 for the validity of this
expansion;
(h) use binomial expansions in approximations.
3
Matrices
16
Candidates should be able to:
3.1
10
(a) identify null, identity, diagonal, triangular and
symmetric matrices;
Matrices
(b) use the conditions for the equality of two
matrices;
(c) perform scalar multiplication, addition,
subtraction and multiplication of matrices with
at most three rows and three columns;
(d) use the properties of matrix operations;
(e) find the inverse of a non-singular matrix using
elementary row operations;
(f)
evaluate the determinant of a matrix;
(g) use the properties of determinants;
3.2
4
Systems of linear
equations
6
(h) reduce an augmented matrix to row-echelon
form, and determine whether a system of linear
equations has a unique solution, infinitely
many solutions or no solution;
(i)
apply the Gaussian elimination to solve a
system of linear equations;
(j)
find the unique solution of a system of linear
equations using the inverse of a matrix.
Differentiation
20
Candidates should be able to:
4.1
6
(a) determine the existence and values of the lefthand limit, right-hand limit and limit of a
function;
Limits
(b) use the properties of limits;
4.2
Derivatives
6
(c) identify the derivative of a function as a limit;
(d) find the derivatives of xn (n ∈ _ ), ex, ln x, with
constant multiples, sums, differences,
products, quotients and composites;
(e) perform implicit differentiation.
3
Topic
4.3
Applications of
differentiation
Teaching
Period
8
Learning Outcome
(f)
determine where a function is increasing,
decreasing, concave upward and concave
downward;
(g) determine the stationary points, extremum
points and points of inflexion;
(h) sketch the graphs of functions, including
asymptotes parallel to the coordinate axes.
5
Integration
20
Candidates should be able to:
5.1
10
(a) identify integration as the reverse of
differentiation;
Indefinite integrals
(b) integrate xn (n ∈ _ ) and ex, with constant
multiples, sums and differences;
(c) integrate rational functions by means of
decomposition into partial fractions;
(d) use algebraic substitutions to find integrals;
(e) perform integration by parts;
5.2
Definite integrals
10
(f)
identify a definite integral as the area under a
curve;
(g) use the properties of definite integrals;
(h) evaluate definite integrals;
6
Differential Equations
14
(i)
calculate the area of a region bounded by a
curve and lines parallel to the coordinate axes,
or between two curves;
(j)
calculate volumes of solids of revolution about
one of the coordinate axes.
Candidates should be able to:
(a) find the general solution of a first order
differential equation with separable variables;
(b) find the general solution of a first order linear
differential equation by means of an integrating
factor;
(c) transform, by a given substitution, a first order
differential equation into one with separable
variables or one which is linear;
(d) use a boundary condition to find a particular
solution;
(e) solve problems, related to social sciences and
management, that can be modelled by
differential equations.
4
SECOND TERM: STATISTICS
Topic
7
Data Description
Teaching
Period
14
Learning Outcome
Candidates should be able to:
(a) identify discrete, continuous, ungrouped and
grouped data;
(b) construct and interpret stem-and-leaf diagrams,
box-and-whisker plots, histograms and
cumulative frequency curves;
(c) state the mode and range of ungrouped data;
(d) determine the median and interquartile range
of ungrouped and grouped data;
(e) calculate the mean and standard deviation of
ungrouped and grouped data from raw data
n
and from given totals such as ∑ ( xi − a ) and
i =1
n
2
∑ ( xi − a ) ;
i =1
(f)
select and use the appropriate measures of
central tendency and measures of dispersion;
(g) calculate the Pearson coefficient of skewness;
(h) describe the shape of a data distribution.
8
Probability
14
Candidates should be able to:
(a) apply the addition principle and the
multiplication principle;
(b) use the formulae for combinations and
permutations in simple cases;
(c) identify a sample space, and calculate the
probability of an event;
(d) identify complementary, exhaustive and
mutually exclusive events;
(e) use the formula
P(A ∪ B) = P(A) + P(B) − P(A ∩ B);
(f)
calculate conditional probabilities, and identify
independent events;
(g) use the formulae
P(A ∩ B) = P(A) × P(B | A) = P(B) × P(A | B);
(h) use the rule of total probability.
5
Topic
9
Teaching
Period
Learning Outcome
Probability Distributions
26
Candidates should be able to:
9.1
6
(a) identify discrete random variables;
Discrete random
variables
(b) construct a probability distribution table for a
discrete random variable;
(c) use the probability function and cumulative
distribution function of a discrete random
variable;
(d) calculate the mean and variance of a discrete
random variable;
9.2
Continuous random
variables
6
(e) identify continuous random variables;
(f)
relate the probability density function and
cumulative distribution function of a
continuous random variable;
(g) use the probability density function and
cumulative distribution function of a
continuous random variable;
(h) calculate the mean and variance of a
continuous random variable;
9.3
9.4
Binomial distribution
Poisson distribution
4
4
(i)
use the probability function of a binomial
distribution, and find its mean and variance;
(j)
use the binomial distribution as a model for
solving problems related to social sciences and
management;
(k) use the probability function of a Poisson
distribution, and identify its mean and
variance;
(l)
9.5
Normal distribution
6
use the Poisson distribution as a model for
solving problems related to social sciences and
management;
(m) identify the general features of a normal
distribution, in relation to its mean and
standard deviation;
(n) standardise a normal random variable, and use
the normal distribution tables;
(o) use the normal distribution as a model for
solving problems related to social sciences and
management;
(p) use the normal distribution, with continuity
correction, as an approximation to the
binomial distribution, where appropriate.
6
Topic
Teaching
Period
Learning Outcome
10 Correlation and
Regression
22
Candidates should be able to:
10.1 Correlation
8
(a) plot a scatter diagram, and describe the
correlation between two variables;
(b) identify a spurious correlation;
(c) calculate and interpret the Pearson correlation
coefficient;
(d) calculate and interpret the Spearman rank
correlation coefficient;
10.2 Regression
14
(e) find the equation of a least squares regression
line;
(f)
interpret regression coefficients;
(g) use a regression line to make predictions;
(h) calculate and interpret the coefficient of
determination;
(i)
11 Index Numbers
12
use transformation of variables to obtain
linearity in regression analysis.
Candidates should be able to:
(a) explain the meaning of an index number;
(b) calculate and interpret the simple average of
relatives;
(c) calculate and interpret the weighted average of
relatives;
(d) calculate and interpret simple aggregate
indices;
(e) calculate and interpret Laspeyres and Paasche
indices.
12 Time Series
20
Candidates should be able to:
(a) identify the components of a time series;
(b) identify an additive and a multiplicative time
series model;
(c) plot and comment on a time series;
(d) use the method of moving averages to smooth
a time series;
(e) calculate and interpret seasonal indices;
(f)
obtain a seasonally adjusted time series;
(g) use the least squares method to fit a trend line;
(h) use a trend line for forecasting.
7
THIRD TERM: FINANCIAL AND DECISION MATHEMATICS
Topic
13 Interest, Annuity and
Depreciation
13.1 Interest
Teaching
Period
Learning Outcome
20
Candidates should be able to:
8
(a) calculate simple interest and compound
interest;
(b) calculate the compound amount involving a
number of interest periods;
(c) determine an effective interest rate;
(d) solve problems related to savings and loans;
13.2 Annuity
8
(e) calculate the future value of an annuity;
(f)
solve problems related to sinking funds;
(g) calculate the present value of an annuity;
(h) solve problems related to amortisation;
13.3 Depreciation
14 Cost, Revenue and Profit
4
16
(i)
use the flat rate depreciation method;
(j)
use the reducing balance depreciation method.
Candidates should be able to:
(a) find cost, revenue and profit functions, and use
their relationship;
(b) determine the break-even point;
(c) find average cost, average revenue and average
profit functions;
(d) find marginal cost, marginal revenue and
marginal profit functions;
(e) determine the minimum cost, maximum
revenue and maximum profit;
(f)
find demand and supply functions;
(g) determine the equilibrium price and
equilibrium demand;
(h) calculate and interpret the consumer’s surplus
and producer’s surplus.
8
Topic
15 Linear Programming
15.1 Problem formulation
Teaching
Period
Learning Outcome
20
Candidates should be able to:
4
(a) determine the constraints;
(b) find an objective function;
(c) state a linear programming model;
15.2 Graphical method
8
(d) identify a feasible region;
(e) determine the optimal solution;
(f)
15.3 Simplex method
8
identify special cases: infeasibility, unbounded
solutions and multiple optimal solutions;
(g) obtain the standard simplex form using slack
and/or surplus variables;
(h) construct simplex tableaux for a maximisation
problem with at most three variables and three
constraints in addition to non-negativity
conditions;
(i)
16 Critical Path Analysis
16.1 Networks
determine the optimal solution.
20
Candidates should be able to:
4
(a) identify activities on arcs and activities on
nodes;
(b) construct an activity network for a project;
16.2 Critical paths
8
(c) calculate the earliest and latest start and finish
times for activities;
(d) calculate and interpret the total float for an
activity;
(e) identify critical activities and critical paths;
(f)
16.3 Scheduling and
crashing
8
determine the minimum completion time of a
project;
(g) construct a Gantt chart and resource histogram
for a project;
(h) determine the minimum number of workers to
complete a project in a given time;
(i)
9
determine the minimum time to complete a
project for a given number of workers;
Topic
Teaching
Period
Learning Outcome
(j)
determine the effect of adjusting the duration
of an activity on the critical path and
completion time;
(k) determine the effect of adjusting the number of
workers on the critical path and completion
time.
17 Inventory Models
16
Candidates should be able to:
(a) identify assumptions of the basic economic
order quantity model;
(b) use the basic economic order quantity model;
(c) use the economic order quantity model with
planned shortages;
(d) find the holding cost and ordering cost;
(e) find the reorder point, cycle time and number
of orders per year;
(f)
determine the minimum total cost order
quantity;
(g) determine the minimum total cost per year.
18 Game Theory
16
Candidates should be able to:
(a) explain the meaning of a two-person zero-sum
game;
(b) determine the outcomes for players from a
given pay-off matrix;
(c) determine play-safe strategies and identify a
stable solution;
(d) find the expected value of a game;
(e) use a dominance argument to reduce a pay-off
matrix;
(f)
determine optimal mixed strategies for a game
with no stable solution;
(g) use a graphical method to solve a game, where
appropriate.
10
Coursework
The Mathematics (M) coursework is intended to enable candidates to carry out mathematical
investigation and mathematical modelling, so as to enhance the understanding of mathematical
processes and applications and to develop soft skills.
The coursework comprises three assignments set down by the Malaysian Examinations Council.
The assignments are based on three different areas of the syllabus and represent two types of tasks:
mathematical investigation and mathematical modelling.
A school candidate is required to carry out one assignment in each term under the supervision of
the subject teacher as specified in the Teacher’s Manual for Mathematics (M) Coursework which can
be downloaded from MEC’s Portal (http://www.mpm.edu.my) by the subject teacher during the first
term of form six. The assignment reports are graded by the subject teacher in the respective terms. A
viva session is conducted by the teacher in each term after the assessment of the assignment reports.
An individual private candidate is required to carry out one assignment in each term as specified
in the Individual Private Candidate’s Manual for Mathematics (M) Coursework which can be
downloaded from MEC’s Portal (http://www.mpm.edu.my) by the candidate during the first term of
form six. The assignment reports are graded by an external examiner in the respective terms. A viva
session is conducted by the examiner in each term after the assessment of the assignment reports.
A repeating candidate may use the total mark obtained in the coursework for the subsequent
STPM examination. Requests to carry forward the moderated coursework mark should be made
during the registration of the examination.
11
Scheme of Assessment
Term of
Study
Paper Code
and Name
First
Term
950/1
Mathematics (M)
Paper 1
Mark
(Weighting)
Type of Test
Duration
Administration
1½ hours
Central assessment
1½ hours
Central assessment
1½ hours
Central assessment
60
(26.67%)
Written test
Section A
Answer all 6 questions
of variable marks.
45
Section B
Answer 1 out of 2
questions.
15
All questions are based
on topics 1 to 6.
Second
Term
950/2
Mathematics (M)
Paper 2
Written test
60
(26.67%)
Section A
Answer all 6 questions
of variable marks.
45
Section B
Answer 1 out of 2
questions.
15
All questions are based
on topics 7 to 12.
Third
Term
950/3
Mathematics (M)
Paper 3
60
(26.67%)
Written test
Section A
Answer all 6 questions
of variable marks.
45
Section B
Answer 1 out of 2
questions.
15
All questions are based
on topics 13 to 18.
First,
Second
and
Third
Terms
950/4
Mathematics (M)
Paper 4
Coursework
3 assignments, each
based on topics 1 to 6,
topics 7 to 12 and topics
13 to 18.
12
180
to be
scaled to
45
(20%)
Throughout
the three
terms
Assessment by school
teachers for candidates
from government and
government-aided
schools
Assessment by
appointed assessors for
candidates from private
schools and individual
private candidates
Performance Descriptions
A grade A candidate is likely able to:
(a)
use correctly mathematical concepts, terminology and notation;
(b)
display and interpret mathematical information in tabular, diagrammatic and graphical
forms;
(c)
identify mathematical patterns and structures in a variety of situations;
(d)
use appropriate mathematical models in different contexts;
(e)
apply correctly mathematical principles and techniques in solving problems;
(f)
carry out calculations and approximations to an appropriate degree of accuracy;
(g)
interpret the significance and reasonableness of results, making sensible predictions where
appropriate;
(h)
present mathematical explanations, arguments and conclusions, usually in a logical and
systematic manner.
A grade C candidate is likely able to:
(a)
use correctly some mathematical concepts, terminology and notation;
(b)
display and interpret some mathematical information in tabular, diagrammatic and graphical
forms;
(c)
identify mathematical patterns and structures in certain situations;
(d)
use appropriate mathematical models in certain contexts;
(e)
apply correctly some mathematical principles and techniques in solving problems;
(f)
carry out some calculations and approximations to an appropriate degree of accuracy;
(g)
interpret the significance and reasonableness of some results;
(h)
present some mathematical explanations, arguments and conclusions.
13
Mathematical Notation
Miscellaneous symbols
=
is equal to
≠
is not equal to
≡
is identical to or is congruent to
≈
is approximately equal to
<
is less than
<
is less than or equal to
>
is greater than
>
is greater than or equal to
∞
infinity
∴
therefore
Operations
a+b
a plus b
a−b
a minus b
a × b, ab
a multiplied by b
a ÷ b,
a
b
a:b
a
a divided by b
ratio of a to b
nth power of a
n
1
2
a ,
1
n
a ,
n
a
positive square root of a
a
positive nth root of a
|a|
absolute value of a real number a
n
∑u
u1 + u2 + ⋅ ⋅ ⋅ + un
n!
n factorial for n ∈ `
⎛n⎞
⎜ ⎟
⎝r⎠
binomial coefficient
i =1
i
n!
for n, r ∈ ` , 0 < r < n
r !( n − r )!
Set notation
∈
is an element of
∉
is not an element of
∅
empty set
{x|...}
set of x such that . . .
`
set of natural numbers, {0, 1, 2, 3, . . .}
]
set of integers
]+
_
set of positive integers
set of rational numbers
\
set of real numbers
14
[ a, b ]
closed interval {x | x ∈ \ , a < x < b}
( a, b )
open interval {x | x ∈ \ , a < x < b}
[ a, b )
interval {x | x ∈ \ , a < x < b}
( a, b ]
∪
interval {x | x ∈ \ , a < x < b}
union
∩
intersection
Functions
f
a function f
f(x)
value of a function f at x
f :A→ B
f :x6 y
f is a function under which each element of set A has an image in set B
f −1
inverse function of f
f Dg
composite function of f and g which is defined by f D g(x) = f [g(x)]
e
exponential function of x
loga x
logarithm to base a of x
ln x
natural logarithm of x, loge x
x
f is a function which maps the element x to the element y
Matrices
A
a matrix A
0
null matrix
identity matrix
I
T
transpose of a matrix A
A
−1
A
inverse of a non-singular square matrix A
det A
determinant of a square matrix A
Derivatives and integrals
lim f ( x)
limit of f(x) as x tends to a
dy
dx
first derivative of y with respect to x
f '( x)
first derivative of f(x) with respect to x
d2 y
dx 2
second derivative of y with respect to x
f ''( x )
second derivative of f(x) with respect to x
dn y
dx n
nth derivative of y with respect to x
f (n ) ( x)
nth derivative of f(x) with respect to x
∫ y dx
indefinite integral of y with respect to x
x →a
∫
b
a
y dx
definite integral of y with respect to x for values of x between a and b
15
Data description
x1, x2, . . .
observations
f1, f2, . . .
frequencies with which the observations x1, x2, . . . occur
x
sample mean
s2
sample variance, s 2 =
μ
population mean
σ2
population variance
1
n
n
∑ ( xi − x )
2
i =1
Probability
A
an event A
A'
complement of an event A or the event not A
P(A)
probability of an event A
P(A | B)
probability of event A given event B
Probability distributions
X
a random variable X
x
value of a random variable X
Z
standardised normal random variable
z
value of the standardised normal random variable Z
f(x)
value of the probability density function of a continuous random variable X
F(x)
value of the cumulative distribution function of a continuous random variable X
E(X)
expectation of a random variable X
Var(X)
variance of a random variable X
B(n, p)
binomial distribution with parameters n and p
Po(λ)
Poisson distribution with parameter λ
N(μ, σ 2 )
normal distribution with mean μ and variance σ 2
Correlation and regression
r
correlation coefficient for a sample
r2
Sxx , Syy
Sxy
coefficient of determination
sums of squared deviations from the means
sum of the cross products of deviations from the means
Interest and annuity
P
principal
I
interest
A
accrued amount
r
nominal interest rate
reff
effective interest rate
t
time in years
n
number of compounding periods
16
i
interest rate per compounding period
A
future value of an annuity
R
payment at the end of each period
P
present value of an annuity
Inventory models
D
demand rate
Ch
holding cost per unit per time period
Cb
backorder cost per unit per time period
Co
ordering cost per order
Q
order quantity
Q*
optimal order quantity
17
Electronic Calculators
During the written paper examination, candidates are advised to have standard scientific calculators
which must be silent. Programmable and graphic display calculators are prohibited.
Reference Books
Teachers and candidates may use books specially written for the STPM examination and other
reference books such as those listed below.
Algebra and Calculus
1.
Harcet, J., Heinrichs, L., Seiler, P.M. and Skoumal, M.T., 2012. Mathematics: Higher
Level, IB Diploma Programme. United Kingdom: Oxford University Press.
2.
Neill, H. and Quadling, D., 2002. Advanced Level Mathematics: Pure Mathematics 1 and 2
& 3. United Kingdom: Cambridge University Press.
3.
Beecher, J.A., Penna, J.A. and Bittinger, M.L., 2012. College Algebra. 4th edition.
Singapore: Pearson Addison-Wesley.
4.
Blitzer, R., 2010. College Algebra: An Early Functions Approach. 2nd edition. Singapore:
Pearson Prentice Hall.
5.
Stewart, J., 2012. Single Variable Calculus: Early Transcendentals. 7th edition, Metric
Version. Singapore: Brooks/Cole, Cengage Learning.
6.
Tan, S.T., 2011. Single Variable Calculus: Early Transcendentals. Singapore: Brooks/Cole,
Cengage Learning.
Statistics
7.
Crawshaw, J. and Chambers, J., 2001. A Concise Course in Advanced Level Statistics.
4th edition. United Kingdom: Nelson Thornes.
8.
Upton, G. and Cook, I., 2001. Introducing Statistics. 2nd edition. United Kingdom: Oxford
University Press.
9.
Curwin, J. and Slater, R., 2008. Quantitative Methods for Business Decisions. 6th edition.
Singapore: South-Western, Cengage Learning.
10.
Francis, A., 2008. Business Mathematics and Statistics. 6th edition. Singapore: SouthWestern, Cengage Learning.
Financial and Decision Mathematics
11.
Lial, M.L., Greenwell, R.N. and Ritchey, N.P., 2012. Finite Mathematics and Calculus with
Applications. 9th edition. Singapore: Pearson Addison-Wesley.
12.
Sullivan, M. and Mizrahi A., 2004. Mathematics: An Applied Approach. 8th edition.
Singapore: John Wiley.
13.
Bloomfield I. and Stevens J., 2002. Complete Advanced Level Mathematics: Discrete and
Decision. United Kingdom: Nelson Thornes.
14.
Anderson, D.R. et al., 2011. An Introduction to Management Science: Quantitative
Approaches to Decision Making. 13th edition. Singapore: South-Western, Cengage
Learning.
18
SPECIMEN PAPER
950/1
STPM
MATHEMATICS (M) (MATEMATIK (M))
PAPER 1 (KERTAS 1)
One and a half hours (Satu jam setengah)
MAJLIS PEPERIKSAAN MALAYSIA
(MALAYSIAN EXAMINATIONS COUNCIL)
SIJIL TINGGI PERSEKOLAHAN MALAYSIA
(MALAYSIA HIGHER SCHOOL CERTIFICATE)
Instruction to candidates:
DO NOT OPEN THIS QUESTION PAPER UNTIL YOU ARE TOLD TO DO SO.
Answer all questions in Section A and any one question in Section B. Answers may be written in
either English or Bahasa Malaysia.
All necessary working should be shown clearly.
Scientific calculators may be used. Programmable and graphic display calculators are
prohibited.
A list of mathematical formulae is provided on page of this question paper.
Arahan kepada calon:
JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU UNTUK BERBUAT
DEMIKIAN.
Jawab semua soalan dalam Bahagian A dan mana-mana satu soalan dalam Bahagian B.
Jawapan boleh ditulis dalam bahasa Inggeris atau Bahasa Malaysia.
Semua kerja yang perlu hendaklah ditunjukkan dengan jelas.
Kalkulator sainstifik boleh digunakan. Kalkulator boleh atur cara dan kalkulator paparan grafik
tidak dibenarkan.
Senarai rumus matematik dibekalkan pada halaman dalam kertas soalan ini.
__________________________________________________________________________________
This question paper consists of
printed pages and
(Kertas soalan ini terdiri daripada halaman bercetak dan
© Majlis Peperiksaan Malaysia
STPM 950/1
19
blank page.
halaman kosong.)
Section A [45 marks]
Answer all questions in this section.
1
The function f is defined by f(x) = ln (1 – 2x), x < 0.
(a) Find f −1, and state its domain.
[3 marks]
−1
2
(b) Sketch, on the same axes, the graphs of f and f .
[4 marks]
(c) Determine whether there is any value of x for which f(x) = f −1(x).
[3 marks]
The sequence u1, u2, u3, . . . is defined by un+1 = 3un, u1 = 2.
(a) Write down the first five terms of the sequence.
[2 marks]
(b) Suggest an explicit formula for ur.
[2 marks]
3 Using an augmented matrix and elementary row operations, find the solution of the system of
equations
3x – 2y – 5z = –5
x + 3y – 2z = –6
5x – 4y + z = 11
[9 marks]
4 Find the gradients of the curve y 3 + y = x3 + x 2 at the points where the curve meets the coordinate
axes.
[6 marks]
5
Show that
e
∫1 x ln x dx = 14 (e
Hence, find the value of
6
e
∫1 x(ln x)
2
2
+ 1).
dx .
[4 marks]
[3 marks]
The variables x and y, where x, y > 0, are related by the differential equation
dy
2y
+ y2 = − .
dx
x
Using the substitution y =
u
, show that the differential equation may be reduced to
x2
du
u2
=− 2.
dx
x
[3 marks]
Solve this differential equation, and hence, find y, in terms of x, with the condition that y = 1
when x = 1.
[6 marks]
950/1
20
Bahagian A [45 markah]
Jawab semua soalan dalam bahagian ini.
1
Fungsi f ditakrifkan oleh f(x) = ln (1 – 2x), x < 0.
(a) Cari f −1, dan nyatakan domainnya.
[3 markah]
−1
2
(b) Lakar, pada paksi yang sama, graf f dan f .
[4 markah]
(c) Tentukan sama ada terdapat sebarang nilai x untuk f(x) = f −1(x).
[3 markah]
Jujukan u1, u2, u3, . . . ditakrifkan oleh un+1 = 3un, u1 = 2.
(a) Tulis lima sebutan yang pertama jujukan itu.
[2 markah]
(b) Cadangkan satu rumus tak tersirat bagi ur.
[2 markah]
3 Dengan menggunakan satu matriks imbuhan dan operasi baris permulaan, cari selesaian sistem
persamaan
3x – 2y – 5z = –5
x + 3y – 2z = –6
5x – 4y + z = 11
[9 markah]
4 Cari kecerunan lengkung y 3 + y = x3 + x 2 di titik di mana lengkung itu bertemu dengan paksi
koordinat.
[6 markah]
5
Tunjukkan bahawa
e
∫1 x ln x dx = 14 (e
Dengan yang demikian, cari nilai
6
e
∫1 x(ln x)
2
2
+ 1).
dx .
[4 markah]
[3 markah]
Pembolahubah x dan y, dengan x, y > 0, dihubungkan oleh persamaan pembezaan
dy
2y
+ y2 = − .
dx
x
Dengan menggunakan gantian y =
u
, tunjukkan bahawa persamaan pembezaan itu boleh
x2
diturunkan kepada
du
u2
=− 2.
dx
x
[3 markah]
Selesaikan persamaan pembezaan ini, dan dengan yang demikian, cari y, dalam sebutan x, dengan
syarat y = 1 apabila x = 1.
[6 markah]
950/1
21
Section B [15 marks]
Answer any one question in this section.
1 + ax
, where | b | < 1, in ascending powers of x up to the term in x3.
1 + bx
Determine the set of values of x for which both the expansions are valid.
[7 marks]
7
2
Expand (1 + x ) 3 and
If the two expansions are identical up to the term in x2,
(a) determine the values of a and b,
(b) use x = 18 to obtain the approximation
[3 marks]
3
81 ≈
212
49
,
[3 marks]
(c) find, correct to five decimal places, the difference between the terms in x3 for the two
[2 marks]
expansions with x = 18 .
8 Sketch, on the same axes, the curve y 2 = x and the straight line y = 2 − x, showing the
coordinates of the points of intersection.
[4 marks]
(a) State whether the curve y 2 = x has a turning point. Justify your answer.
[2 marks]
(b) Calculate the area of the region bounded by the curve y 2 = x and the straight line y = 2 − x.
[4 marks]
(c) Calculate the volume of the solid formed by revolving the region bounded by the
[5 marks]
curve y 2 = x and the straight line y = 2 − x completely about the y-axis.
950/1
22
Bahagian B [15 markah]
Jawab mana-mana satu soalan dalam bahagian ini.
1 + ax
, dengan | b | < 1, dalam kuasa x menaik hingga sebutan dalam x3.
1 + bx
Tentukan set nilai x supaya kedua-dua kembangan itu sah.
[7 markah]
7
2
Kembangkan (1 + x ) 3 dan
Jika dua kembangan itu secaman hingga sebutan dalam x2,
(a) tentukan nilai a dan b,
[3 markah]
(b) gunakan x = 18 untuk memperoleh penghampiran
3
81 ≈
212
49
,
[3 markah]
(c) cari, betul hingga lima tempat perpuluhan, beza antara sebutan dalam x3 bagi dua kembangan
[2 markah]
itu dengan x = 18 .
8 Lakar, pada paksi yang sama, lengkung y 2 = x dan garis lurus y = 2 − x, dengan menunjukkan
koordinat titik persilangan.
[4 markah]
(a) Nyatakan sama ada lengkung y 2 = x mempunyai titik pusingan. Justifikasikan jawapan anda.
[2 markah]
(b) Hitung luas rantau yang dibatasi oleh lengkung y 2 = x dan garis lurus y = 2 − x.
[4 markah]
(c) Hitung isipadu pepejal yang terbentuk dengan mengisar rantau yang dibatasi oleh
lengkung y 2 = x dan garis lurus y = 2 − x selengkapnya di sekitar paksi-y.
[5 markah]
950/1
23
MATHEMATICAL FORMULAE (RUMUS MATEMATIK)
Binomial expansions (Kembangan binomial)
⎛n⎞
⎝1⎠
⎛n⎞
⎝2⎠
⎛n⎞
⎝r⎠
(a + b)n = a n + ⎜ ⎟ a n −1b + ⎜ ⎟ a n − 2 b 2 + ⋅ ⋅ ⋅ + ⎜ ⎟ a n − r b r + ⋅ ⋅ ⋅ + b n , n ∈ ] +
(1 + x)n = 1 + nx +
n( n − 1)
2!
x +⋅ ⋅ ⋅+
2
n( n − 1). . .( n − r + 1)
r!
Integration (Pengamiran)
f '( x)
∫ f ( x) dx
= ln |f(x)| + c
dv
du
∫ u dxdx = uv − ∫ v dx dx
950/1
24
x + ⋅ ⋅ ⋅, n ∈ _ , |x| < 1
r
SPECIMEN PAPER
950/2
STPM
MATHEMATICS (M) (MATEMATIK (M))
PAPER 2 (KERTAS 2)
One and a half hours (Satu jam setengah)
MAJLIS PEPERIKSAAN MALAYSIA
(MALAYSIAN EXAMINATIONS COUNCIL)
SIJIL TINGGI PERSEKOLAHAN MALAYSIA
(MALAYSIA HIGHER SCHOOL CERTIFICATE)
Instruction to candidates:
DO NOT OPEN THIS QUESTION PAPER UNTIL YOU ARE TOLD TO DO SO.
Answer all questions in Section A and any one question in Section B. Answers may be written in
either English or Bahasa Malaysia.
All necessary working should be shown clearly.
Scientific calculators may be used. Programmable and graphic display calculators are
prohibited.
A list of mathematical formulae, statistical tables and graph papers are provided on pages
this question paper.
of
Arahan kepada calon:
JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU UNTUK BERBUAT
DEMIKIAN.
Jawab semua soalan dalam Bahagian A dan mana-mana satu soalan dalam Bahagian B.
Jawapan boleh ditulis dalam bahasa Inggeris atau Bahasa Malaysia.
Semua kerja yang perlu hendaklah ditunjukkan dengan jelas.
Kalkulator sainstifik boleh digunakan. Kalkulator boleh atur cara dan kalkulator paparan grafik
tidak dibenarkan.
Senarai rumus matematik, jadual statistik dan kertas graf dibekalkan pada halaman dalam
kertas soalan ini.
__________________________________________________________________________________
This question paper consists of
printed pages and
(Kertas soalan ini terdiri daripada halaman bercetak dan
© Majlis Peperiksaan Malaysia
STPM 950/2
25
blank page.
halaman kosong.)
Section A [45 marks]
Answer all questions in this section.
1 The lengths (in seconds) of 60 songs recorded by a certain group of singers are summarised in the
table below.
Song length (x)
Number of songs
0 < x < 120
1
120 < x < 180
9
180 < x < 240
15
240 < x < 300
17
300 < x < 360
13
360 < x < 600
5
(a) Display the data on a histogram.
[3 marks]
(b) Calculate the mean song length to the nearest half second.
[2 marks]
(c) Calculate the standard deviation of the song lengths to the nearest half second.
[3 marks]
2 Two companies X and Y operate call-taxi services in a town. The percentages of residents in the
town using the taxi services from companies X and Y are 40% and 60% respectively. The
probabilities of taxis from companies X and Y being late are 0.02 and 0.01 respectively. A taxi is
booked at random. Find the probability that
3
(a) the taxi is late and it is from company X,
[2 marks]
(b) the taxi is late.
[3 marks]
The probability density function of a continuous random variable X is given by
⎧ ax(b − x),
0 < x < b,
⎩0,
otherwise,
f ( x) = ⎨
where a and b are positive constants. It has a value of
1
2
at the mean value of X.
(a) Determine the values of a and b.
[7 marks]
(b) Sketch the graph of the probability density function.
[2 marks]
950/2
26
Bahagian A [45 markah]
Jawab semua soalan dalam bahagian ini.
1 Panjang (dalam saat) 60 lagu yang dirakamkan oleh sekumpulan penyanyi diiktisarkan dalam
jadual di bawah.
Panjang lagu (x)
Bilangan lagu
0 < x < 120
1
120 < x < 180
9
180 < x < 240
15
240 < x < 300
17
300 < x < 360
13
360 < x < 600
5
(a) Paparkan data itu pada satu histogram.
[3 markah]
(b) Hitung min panjang lagu hingga setengah saat terdekat.
[2 markah]
(c) Hitung sisihan piawai panjang lagu itu hingga setengah saat terdekat.
[3 markah]
2 Dua syarikat X dan Y mengusahakan perkhidmatan teksi panggilan di sebuah bandar. Peratusan
penduduk di bandar itu yang menggunakan perkhidmatan teksi dari syarikat X dan Y masing-masing
ialah 40% dan 60%. Kebarangkalian teksi dari syarikat X dan Y lewat masing-masing ialah 0.02 dan
0.01. Satu teksi ditempah secara rawak. Cari kebarangkalian bahawa
3
(a) teksi itu lewat dan teksi itu adalah dari syarikat X,
[2 markah]
(b) teksi itu lewat.
[3 markah]
Fungsi ketumpatan kebarangkalian pembolehubah rawak selanjar X diberikan oleh
⎧ ax(b − x),
f ( x) = ⎨
⎩0,
0 < x < b,
jika tidak,
dengan a dan b pemalar positif. Fungsi itu mempunyai nilai
1
2
pada min nilai X.
(a) Tentukan nilai a dan b.
[7 markah]
(b) Lakar graf fungsi ketumpatan kebarangkalian itu.
[2 markah]
950/2
27
4 A study is conducted to assess the impact of the size of a stall, x (in m2) on daily sales, y (in RM).
A random sample of six stalls is taken from several shopping centres. The data obtained are
summarised as follows:
∑ x = 24 400, ∑ y = 28 368, ∑ ( x − x )( y − y ) = 6780,
∑ (x − x )
2
= 186 333,∑ ( y − y ) 2 = 130 110.
[4 marks]
Calculate the coefficient of determination, and comment on your answer.
The study also assesses the impact of the size of a shopping centre on daily sales and finds that
the coefficient of determination is 0.674. State whether the size of a stall or the size of a shopping
centre is more suitable to be used to predict daily sales. Give a reason for your answer.
[2 marks]
5 The average prices (in thousand ringgit) and the daily quantities for three models of cars produced
by a car manufacturer for years 2004 to 2006 are given as follows:
Year
Model A
Model B
Model C
Price
Quantity
Price
Quantity
Price
Quantity
2004
62
120
46
160
38
100
2005
67
200
47
180
33
90
2006
59
140
48
200
29
70
(a) Taking year 2004 as the base year, calculate the producer price indices for years 2005 and
2006.
[4 marks]
(b) Comment on the producer price indices obtained.
[1 mark]
6 The table below shows the quarterly water consumption (in thousand cubic metres) of a factory
from the year 2004 to 2007.
Year
Water consumption
Quarter 1
Quarter 2
Quarter 3
Quarter 4
2004
72
56
50
69
2005
73
56
49
70
2006
76
57
50
71
2007
77
58
52
72
(a) Plot a time series, and comment on the appropriateness of a linear trend.
[4 marks]
(b) Calculate the centred four-quarter moving averages for this time series.
[4 marks]
(c) Calculate the seasonal indices using an additive model.
[4 marks]
950/2
28
4 Satu kajian dijalankan untuk menaksir impak saiz gerai, x (dalam m2) terhadap jualan harian, y
(dalam RM). Satu sampel rawak enam gerai diambil daripada beberapa pusat beli-belah. Data yang
diperoleh adalah diikhtisarkan seperti yang berikut:
∑ x = 24 400, ∑ y = 28 368, ∑ ( x − x )( y − y ) = 6780,
∑ (x − x )
2
= 186 333,∑ ( y − y ) 2 = 130 110.
[4 markah]
Hitung pekali penentuan, dan berikan ulasan tentang jawapan anda.
Kajian itu juga menaksir impak saiz pusat beli-belah terhadap jualan harian dan mendapati
bahawa pekali penentuan ialah 0.674. Nyatakan sama ada saiz gerai atau saiz pusat beli-belah lebih
sesuai digunakan untuk meramalkan jualan harian. Berikan satu sebab bagi jawapan anda. [2 markah]
5 Harga purata (dalam ribu ringgit) dan kuantiti harian bagi tiga model kereta yang dihasilkan oleh
sebuah pengilang kereta bagi tahun 2004 hingga 2006 diberikan seperti yang berikut:
Tahun
Model A
Model B
Model C
Harga
Kuantiti
Harga
Kuantiti
Harga
Kuantiti
2004
62
120
46
160
38
100
2005
67
200
47
180
33
90
2006
59
140
48
200
29
70
(a) Dengan mengambil tahun 2004 sebagai tahun asas, hitung indeks harga pengeluar bagi tahun
2005 dan 2006.
[4 markah]
(b) Berikan ulasan tentang indeks harga pengeluar yang diperoleh.
[1 markah]
6 Jadual dibawah menunjukkan penggunaan air (dalam ribu meter padu) suku tahunan sebuah
kilang dari tahun 2004 hingga 2007.
Tahun
Penggunaan air
Sukuan 1
Sukuan 2
Sukuan 3
Sukuan 4
2004
72
56
50
69
2005
73
56
49
70
2006
76
57
50
71
2007
77
58
52
72
(a) Plot satu siri masa, dan berikan ulasan tentang kesesuaian trend linear.
[4 markah]
(b) Hitung purata bergerak empat-sukuan berpusat bagi siri masa ini.
[4 markah]
(c) Hitung indeks bermusim dengan menggunakan satu model berdaya tambah.
[4 markah]
950/2
29
Section B [15 marks]
Answer any one question in this section.
7 An insurance company finds that 8% of the people who are insured with the company against a
certain risk make claims within a year.
(a) Calculate the probability that at least one out of two randomly selected people who are
insured against the risk makes a claim within a year.
[3 marks]
(b) Calculate the probability that at least 3 out of 60 randomly selected people who are insured
against the risk make claims within a year. Comment on the validity of the probability model you use.
[6 marks]
(c) Using an approximate probability distribution, estimate the probability that at least 12 out of
100 randomly selected people who are insured against the risk make claims within a year. Justify your
use of this approximation.
[6 marks]
8 The table below shows the values of the variable y corresponding to eight accurately specified
values of the variable x.
x
5
7
9
11
13
15
17
19
y
8.5
15.0
27.5
35.0
28.0
37.0
46.0
37.0
(a) Plot a scatter diagram for the data.
[2 marks]
(b) Calculate the Pearson correlation coefficient r between x and y.
[5 marks]
(c) Comment on your value of r with respect to the scatter diagram in (a).
[1 mark]
(d) Find the equation of the least squares regression line in the form y = a + bx, where a and b are
constants.
[4 marks]
(e) Estimate the values of y for x = 0, 10, 20, 30, where appropriate. Give a reason why it is
inappropriate to estimate the values of y for the other given values of x.
[3 marks]
950/2
30
Bahagian B [15 markah]
Jawab mana-mana satu soalan dalam bahagian ini.
7 Satu syarikat insurans mendapati bahawa 8% orang yang membeli insurans dengan syarikat ini
terhadap risiko tertentu membuat tuntutan dalam jangka masa satu tahun.
(a) Hitung kebarangkalian bahawa sekurang-kurangnya seorang daripada dua orang yang dipilih
secara rawak yang diinsurankan terhadap risiko itu membuat tuntutan dalam jangka masa satu tahun.
[3 markah]
(b) Hitung kebarangkalian bahawa sekurang-kurangnya tiga orang daripada 60 orang yang dipilih
secara rawak yang diinsurankan terhadap risiko itu membuat tuntutan dalam jangka masa satu
tahun. Berikan ulasan tentang kesahan model kebarangkalian yang anda gunakan.
[6 markah]
(c) Dengan menggunakan taburan kebarangkalian hampiran, anggar kebarangkalian bahawa
sekurang-kurangnya 12 orang daripada 100 orang yang dipilih secara rawak yang diinsurankan
terhadap risiko itu membuat tuntutan dalam jangka masa satu tahun. Justifikasikan penghampiran
yang anda gunakan ini.
[6 markah]
8 Jadual di bawah menunjukkan nilai pembolehubah y yang sepadan dengan lapan nilai
pembolehubah x yang ditentukan secara jitu.
x
5
7
9
11
13
15
17
19
y
8.5
15.0
27.5
35.0
28.0
37.0
46.0
37.0
(a) Plot satu gambar rajah serakan bagi data itu.
[2 markah]
(b) Hitung pekali korelasi Pearson r antara x dengan y.
[5 markah]
(c) Berikan ulasan tentang nilai r terhadap gambar rajah serakan dalam (a).
[1 markah]
(d) Cari persamaan garis regresi kuasa dua terkecil dalam bentuk y = a + bx, dengan a dan b
pemalar.
[4 markah]
(e) Anggar nilai y bagi x = 0, 10, 20, 30, di mana sesuai. Berikan satu sebab mengapa tidak sesuai
[3 markah]
untuk menganggar nilai y bagi nilai x lain yang diberikan.
950/2
31
MATHEMATICAL FORMULAE (RUMUS MATEMATIK)
Probability distributions (Taburan kebarangkalian)
Binomial distribution (Taburan binomial)
⎛n⎞
P(X = x) = ⎜ ⎟ p x (1 − p ) n − x , x = 0, 1, 2,. . ., n
⎝ x⎠
Poisson distribution (Taburan Poisson)
e−λ λ x
, x = 0, 1, 2,. . .
x!
P(X = x) =
Correlation and regression (Korelasi dan regresi)
Pearson correlation coefficient (Pekali korelasi Pearson)
∑ xy −
r=
∑ x2 −
( ∑ x )( ∑ y )
n
(∑ x)
2
∑ y2 −
n
( ∑ y )2
n
Spearman rank correlation coefficient (Pekali korelasi pangkat Spearman)
n
6
r = 1−
∑d
2
i
i =1
n( n − 1)
2
Least squares regression line (Garis regresi kuasa dua terkecil)
y = a + bx , b =
∑ xy −
( ∑ x )( ∑ y )
∑ x2 −
950/2
32
n
( ∑ x )2
n
, a = y − bx
THE NORMAL DISTRIBUTION FUNCTION (FUNGSI TABURAN NORMAL)
If Z has a normal distribution
with mean 0 and variance 1,
then for each value of z, the
tabulated value of Φ(z) is such
that Φ(z) = P(Z < z).
For
negative values of z, use
Φ(−z) = 1 − Φ(z).
Jika Z mempunyai taburan normal
dengan min 0 dan varians 1, maka
bagi setiap nilai z, nilai terjadual
Φ(z)
adalah
sebegitu
rupa
sehinggakan Φ(z) = P(Z < z). Bagi
nilai
negatif
z,
gunakan
Φ(−z) = 1 − Φ(z).
Φ(z)
0
z
z
0
1
2
3
4
5
6
7
8
9
1
2
3 4 5 6 7
ADD (TAMBAH)
8
9
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
0.5000
0.5398
0.5793
0.6179
0.6554
0.6915
0.7257
0.7580
0.7881
0.8159
0.8413
0.8643
0.8849
0.9032
0.9192
0.9332
0.9452
0.9554
0.9641
0.9713
0.9772
0.9821
0.9861
0.9893
0.9918
0.9938
0.9953
0.9965
0.9974
0.9981
0.5040
0.5438
0.5832
0.6217
0.6591
0.6950
0.7291
0.7611
0.7910
0.8186
0.8438
0.8665
0.8869
0.9049
0.9207
0.9345
0.9463
0.9564
0.9649
0.9719
0.9778
0.9826
0.9864
0.9896
0.9920
0.9940
0.9955
0.9966
0.9975
0.9982
0.5080
0.5478
0.5871
0.6255
0.6628
0.6985
0.7324
0.7642
0.7939
0.8212
0.8461
0.8686
0.8888
0.9066
0.9222
0.9357
0.9474
0.9573
0.9656
0.9726
0.9783
0.9830
0.9868
0.9898
0.9922
0.9941
0.9956
0.9967
0.9976
0.9982
0.5120
0.5517
0.5910
0.6293
0.6664
0.7019
0.7357
0.7673
0.7967
0.8238
0.8485
0.8708
0.8907
0.9082
0.9236
0.9370
0.9484
0.9582
0.9664
0.9732
0.9788
0.9834
0.9871
0.9901
0.9925
0.9943
0.9957
0.9968
0.9977
0.9983
0.5160
0.5557
0.5948
0.6331
0.6700
0.7054
0.7389
0.7704
0.7995
0.8264
0.8508
0.8729
0.8925
0.9099
0.9251
0.9382
0.9495
0.9591
0.9671
0.9738
0.9793
0.9838
0.9875
0.9904
0.9927
0.9945
0.9959
0.9969
0.9977
0.9984
0.5199
0.5596
0.5987
0.6368
0.6736
0.7088
0.7422
0.7734
0.8023
0.8289
0.8531
0.8749
0.8944
0.9115
0.9265
0.9394
0.9505
0.9599
0.9678
0.9744
0.9798
0.9842
0.9878
0.9906
0.9929
0.9946
0.9960
0.9970
0.9978
0.9984
0.5239
0.5636
0.6026
0.6406
0.6772
0.7123
0.7454
0.7764
0.8051
0.8315
0.8554
0.8770
0.8962
0.9131
0.9279
0.9406
0.9515
0.9608
0.9686
0.9750
0.9803
0.9846
0.9881
0.9909
0.9931
0.9948
0.9961
0.9971
0.9979
0.9985
0.5279
0.5675
0.6064
0.6443
0.6808
0.7157
0.7486
0.7794
0.8078
0.8340
0.8577
0.8790
0.8980
0.9147
0.9292
0.9418
0.9525
0.9616
0.9693
0.9756
0.9808
0.9850
0.9884
0.9911
0.9932
0.9949
0.9962
0.9972
0.9979
0.9985
0.5319
0.5714
0.6103
0.6480
0.6844
0.7190
0.7517
0.7823
0.8106
0.8365
0.8599
0.8810
0.8997
0.9162
0.9306
0.9429
0.9535
0.9625
0.9699
0.9761
0.9812
0.9854
0.9887
0.9913
0.9934
0.9951
0.9963
0.9973
0.9980
0.9986
0.5359
0.5753
0.6141
0.6517
0.6879
0.7224
0.7549
0.7852
0.8133
0.8389
0.8621
0.8830
0.9015
0.9177
0.9319
0.9441
0.9545
0.9633
0.9706
0.9767
0.9817
0.9857
0.9890
0.9916
0.9936
0.9952
0.9964
0.9974
0.9981
0.9986
4
4
4
4
4
3
3
3
3
3
2
2
2
2
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
8
8
8
8
7
7
6
6
6
5
5
4
4
3
3
2
2
2
2
1
1
1
1
1
0
0
0
0
0
0
12
12
12
11
11
10
10
9
9
8
7
6
5
5
4
4
3
3
2
2
1
1
1
1
1
0
0
0
0
0
32
31
31
30
29
27
26
24
22
20
18
16
15
13
11
10
8
7
6
5
4
3
3
2
2
1
1
1
1
0
36
35
35
34
32
31
29
27
25
23
21
19
16
14
13
11
9
8
6
5
4
4
3
2
2
1
1
1
1
0
16
16
15
15
14
14
13
12
11
10
9
8
7
6
6
5
4
3
3
2
2
2
1
1
1
1
0
0
0
0
20
20
19
19
18
17
16
15
14
13
11
10
9
8
7
6
5
4
4
3
2
2
2
1
1
1
1
0
0
0
24
24
23
23
22
21
19
18
17
15
14
12
11
10
8
7
6
5
4
4
3
2
2
2
1
1
1
1
0
0
28
28
27
26
25
24
23
21
19
18
16
14
13
11
10
8
7
6
5
4
3
3
2
2
1
1
1
1
0
0
Critical values for the normal distribution (Nilai genting bagi taburan normal)
If Z has a normal distribution with mean 0 and Jika Z mempunyai taburan normal dengan min 0 dan
variance 1, then for each value of p, the varians 1, maka bagi setiap nilai p, nilai terjadual z adalah
tabulated value of z is such that P(Z < z) = p.
sebegitu rupa sehinggakan P(Z < z) = p.
p
0.75
0.9
0.95
0.975
0.99
0.995
0.9975
0.999
0.9995
z
0.674
1.282
1.645
1.960
2.326
2.576
2.807
3.090
3.291
950/2
33
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950/2
34
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38
SPECIMEN PAPER
950/3
STPM
MATHEMATICS (M) (MATEMATIK (M))
PAPER 3 (KERTAS 3)
One and a half hours (Satu jam setengah)
MAJLIS PEPERIKSAAN MALAYSIA
(MALAYSIAN EXAMINATIONS COUNCIL)
SIJIL TINGGI PERSEKOLAHAN MALAYSIA
(MALAYSIA HIGHER SCHOOL CERTIFICATE)
Instruction to candidates:
DO NOT OPEN THIS QUESTION PAPER UNTIL YOU ARE TOLD TO DO SO.
Answer all questions in Section A and any one question in Section B. Answers may be written in
either English or Bahasa Malaysia.
All necessary working should be shown clearly.
Scientific calculators may be used. Programmable and graphic display calculators are
prohibited.
A list of mathematical formulae and graph papers are provided on pages of this question paper.
Arahan kepada calon:
JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU UNTUK BERBUAT
DEMIKIAN.
Jawab semua soalan dalam Bahagian A dan mana-mana satu soalan dalam Bahagian B.
Jawapan boleh ditulis dalam bahasa Inggeris atau Bahasa Malaysia.
Semua kerja yang perlu hendaklah ditunjukkan dengan jelas.
Kalkulator sainstifik boleh digunakan. Kalkulator boleh atur cara dan kalkulator paparan grafik
tidak dibenarkan.
Senarai rumus matematik dan kertas graf dibekalkan pada halaman dalam kertas soalan ini.
__________________________________________________________________________________
This question paper consists of
printed pages and
(Kertas soalan ini terdiri daripada halaman bercetak dan
© Majlis Peperiksaan Malaysia
STPM 950/3
39
blank page.
halaman kosong.)
Section A [45 marks]
Answer all questions in this section.
1 A car which costs RM80 000 is bought with a down payment of RM8000 and a loan of
RM72 000. The loan is amortised with 60 monthly payments at an annual interest rate of 3.6%,
compounded monthly. The car depreciates at the rate of 10% per year.
(a) Calculate the monthly payment needed to amortise the loan.
[3 marks]
(b) Calculate the total amount of interest paid over 5 years.
[2 marks]
(c) Using the reducing balance depreciation method, find the book value of the car when the loan
is paid off.
[3 marks]
2 The demand for a campus newspaper is 2300 copies each week if the newspaper is free of charge;
it drops to 1300 copies each week if the newspaper cost RM0.25 per copy. The university is prepared
to supply 500 copies each week free of charge or alternatively 2000 copies each week at RM0.30 per
copy.
(a) Find the linear demand and supply functions.
[4 marks]
(b) Determine the price at which the newspaper is to be sold so that there is neither a shortage nor
a surplus.
[2 marks]
3 A company produces two types of lamps, A and B, which are made of three types of material: iron
frame, electrical component and plastic component. Each lamp A requires 1 unit of iron frame,
2 units of electrical component and 3 units of plastic component, whereas each lamp B requires 3
units of iron frame, 2 units of electrical component and 1 unit of plastic component. The company has
300 000 units of iron frame, 300 000 units of electrical component and 400 000 units of plastic
component in stock. The profits made from each lamp A and lamp B are RM15.00 and RM20.00
respectively.
(a) Formulate a linear programming problem to maximise profit subject to the constraints.
[4 marks]
(b) Using the graphical method, determine the number of lamp A and the number of lamp B
which give the maximum profit, and find this maximum profit.
[8 marks]
4 A project on setting a student registration system of a college involves seven activities. The
activities, their preceding activities and duration (in days) are listed as follows:
Activity
Preceding activities
Duration
A
B
C
D
E
F
G
−
−
4
2
3
8
6
3
4
−
A
B
C
D,E
Determine the earliest start time and the latest finish time for each activity, and hence, find the
minimum duration for the project to be completed.
[6 marks]
950/3
40
Bahagian A [45 markah]
Jawab semua soalan dalam bahagian ini.
1 Sebuah kereta berharga RM80 000 dibeli dengan bayaran pendahuluan RM8000 dan pinjaman
RM72 000. Pinjaman itu dilunaskan dengan bayaran bulanan 60 bulan pada kadar faedah tahunan
3.6%, dikompaun secara bulanan. Nilai kereta itu susut pada kadar 10% per tahun.
(a) Hitung bayaran balik bulanan yang diperlukan untuk melunaskan pinjaman itu.
[3 markah]
(b) Hitung jumlah amaun faedah yang dibayar selama 5 tahun.
[2 markah]
(c) Dengan menggunakan kaedah susut nilai baki berkurangan, cari nilai buku kereta itu apabila
pinjaman habis diselesaikan.
[3 markah]
2 Permintaan akhbar kampus ialah 2300 naskhah setiap minggu jika akhbar itu percuma;
permintaan itu menurun kepada 1300 naskhah setiap minggu jika harga akhbar itu ialah RM0.25 per
naskhah. Universiti itu bersedia untuk menawarkan 500 naskhah percuma setiap minggu atau sebagai
alternatif 2000 naskhah setiap minggu dengan harga RM0.30 per naskhah.
(a) Cari fungsi permintaan dan fungsi penawaran linear.
[4 markah]
(b) Tentukan harga surat khabar yang perlu dijual supaya tidak berlaku kurangan atau lebihan.
[2 markah]
3 Satu syarikat menghasilkan dua jenis lampu, A dan B, yang diperbuat daripada tiga jenis bahan:
bingkai besi, komponen elektrik dan komponen plastik. Setiap lampu A memerlukan 1 unit bingkai
besi, 2 unit komponen elektrik dan 3 unit komponen plastik, manakala setiap lampu B memerlukan
3 unit bingkai besi, 2 unit komponen elektrik dan 1 unit komponen plastik. Syarikat itu mempunyai
300 000 unit bingkai besi, 300 000 unit komponen elektrik dan 400 000 unit komponen plastik dalam
stok. Keuntungan yang diperoleh daripada setiap lampu A dan lampu B masing-masing ialah
RM15.00 dan RM20.00.
(a) Rumuskan satu masalah pengaturcaraan linear untuk memaksimumkan keuntungan tertakluk
kepada kekangan itu.
[4 markah]
(b) Dengan menggunakan kaedah bergraf, tentukan bilangan lampu A dan bilangan lampu B yang
memberikan keuntungan maksimum, dan cari keuntungan maksimum ini.
[8 markah]
4 Satu projek tentang pembangunan sistem pendaftaran pelajar di sebuah kolej melibatkan tujuh
kegiatan. Kegiatan, kegiatan sebelumnya dan tempoh (dalam hari) disenaraikan seperti yang berikut:
Kegiatan
Kegiatan sebelumnya
Tempoh
A
B
C
D
E
F
G
−
−
4
2
3
8
6
3
4
−
A
B
C
D,E
Tentukan masa mula terawal dan masa tamat terlewat bagi setiap kegiatan, dan dengan yang
demikian, cari tempoh minimum bagi menyempurnakan projek itu.
[6 markah]
950/3
41
5 A retailer expects to sell 500 units of an electronic component per year. The cost per unit is RM30
and the ordering cost is RM50 per order. The holding cost per unit of this product is RM3 per year.
The replacement is instantaneous and no shortages are allowed.
6
(a) Find the optimum economic lot size.
[2 marks]
(b) Determine the number of orders per year.
[3 marks]
(c) Calculate the total cost per year.
[4 marks]
A two-person zero-sum game for players A and B is represented by the pay-off matrix
B
A
I
II
III
I
2
1
3
II
3
−5
4
III
0
1
5
(a) Show that the game has a stable solution.
[3 marks]
(b) State the value of the game.
[1 mark]
950/3
42
5 Seorang peruncit menjangka untuk menjual 500 unit sejenis komponen elektronik per tahun. Kos
per unit ialah RM30 dan kos pesanan ialah RM50 per pesanan. Kos pegangan per unit produk ini ialah
RM3 per tahun. Penggantian adalah serta merta dan tiada kurangan yang dibenarkan.
6
(a) Cari saiz lot ekonomi optimum.
[2 markah]
(b) Tentukan bilangan pesanan per tahun.
[3 markah]
(c) Hitung jumlah kos per tahun.
[4 markah]
Satu permainan two-person zero-sum bagi pemain A dan B diwakilkan oleh matriks timbal-balik
B
A
I
II
III
I
2
1
3
II
3
−5
4
III
0
1
5
(a) Tunjukkan bahawa permainan itu mempunyai satu penyelesaian stabil.
[3 markah]
(b) Nyatakan nilai permainan itu.
[1 markah]
950/3
43
Section B [15 marks]
Answer any one question in this section.
7 The number of units, x, demanded for a certain product depends on the unit price, p (in RM), and
is given by
x = 120 000 − 2 p ,
where 0 < p < 60 000. The cost C (in RM) of producing x units is given by
C = 35 000 x − 0.1x 3 ,
where 0 < x < 300.
(a) Find the revenue in terms of x.
[3 marks]
(b) Determine the demand and the unit price for which the revenue is maximum and find the
maximum revenue.
[6 marks]
(c) Calculate the average cost per unit at the production level in (b).
[2 marks]
(d) Determine the production levels at which the profit is increasing.
[4 marks]
8 Two computer companies X and Y are the only sale and service providers in a region. Both
companies advertise their sales and services through radio (I), television (II) and newspaper (III).
Depending on the effectiveness of each advertising campaign, a company can capture a portion of the
customers from the other. The following pay-off matrix summarises the percentages of the customers
captured or lost by company X.
Y
X
I
II
III
I
5
–15
5
II
–5
10
–20
III
15
5
–10
(a) State a reason why company Y should not use the radio as a medium of advertising. [1 mark]
(b) Use a graphical method to determine the optimal strategy of company Y, and hence, find the
percentage gain of the company.
[10 marks]
(c) Determine the optimal strategy of company X.
950/3
44
[4 marks]
Bahagian B [15 markah]
Jawab mana-mana satu soalan dalam bahagian ini.
7 Bilangan unit, x, yang diminta bagi satu produk tertentu bergantung pada harga unit, p
(dalam RM), dan diberikan oleh
x = 120 000 − 2 p ,
dengan 0 < p < 60 000. Kos C (dalam RM) untuk menghasilkan x unit diberikan oleh
C = 35 000 x − 0.1x 3 ,
dengan 0 < x < 300.
(a) Cari hasil dalam sebutan x.
itu.
[3 markah]
(b) Tentukan permintaan dan harga unit supaya hasil adalah maksimum dan cari hasil maksimum
[6 markah]
(c) Hitung kos purata per unit pada aras pengeluaran dalam (b).
[2 markah]
(d) Tentukan aras pengeluaran supaya keuntungan menokok.
[4 markah]
8 Dua syarikat komputer X dan Y sahaja yang menawarkan jualan dan perkhidmatan di satu rantau.
Kedua-dua syarikat mengiklankan jualan dan perkhidmatan mereka melalui radio (I), televisyen (II),
dan surat khabar (III). Bergantung kepada keberkesanan setiap kempen pengiklanan, satu syarikat
boleh menarik sebahagian daripada pelanggan daripada syarikat pesaing. Matriks timbal-balik yang
berikut mengikhtisarkan peratusan pelanggan yang ditarik atau hilang oleh syarikat X.
Y
X
I
II
III
I
5
–15
5
II
–5
10
–20
III
15
5
–10
(a) Nyatakan satu sebab mengapa syarikat Y tidak patut menggunakan radio sebagai perantara
pengiklanan.
[1 markah]
(b) Gunakan kaedah bergraf untuk menentukan strategi optimum syarikat Y, dan dengan yang
demikian, cari peratusan keuntungan syarikat itu.
[10 markah]
(c) Tentukan strategi optimum syarikat X.
[4 markah]
950/3
45
MATHEMATICAL FORMULAE (RUMUS MATEMATIK)
Interest and annuity (Faedah dan anuiti)
Effective interest rate (Kadar faedah berkesan)
reff = (1 + i ) − 1
n
Future value of an annuity (Nilai hadapan anuiti)
⎡ (1 + i )n − 1 ⎤
A= R⎢
⎥
i
⎢⎣
⎥⎦
Present value of an annuity (Nilai kini anuiti)
⎡ 1 − (1 + i )− n ⎤
P = R⎢
⎥
i
⎢⎣
⎥⎦
Inventory models (Model inventori)
For basic EOQ model (Bagi model EOQ asas)
2 DC0
Q* =
Ch
For EOQ model with planned shortages (Bagi model EOQ dengan kurangan terancang)
Q* =
2 DC0
Ch
950/3
46
⎛ C h + Cb ⎞
⎜ C ⎟
⎝
⎠
b
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47
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950/3
48
SPECIMEN ASSIGNMENT
950/4
STPM
MATHEMATICS (M) (MATEMATIK (M))
PAPER 4 (KERTAS 4)
MAJLIS PEPERIKSAAN MALAYSIA
(MALAYSIAN EXAMINATIONS COUNCIL)
SIJIL TINGGI PERSEKOLAHAN MALAYSIA
(MALAYSIA HIGHER SCHOOL CERTIFICATE)
© Majlis Peperiksaan Malaysia
STPM 950/4
49
Many projects require proper planning, management and resource scheduling in order that they can be
completed in time. A project is modelled by a network.
The first step in the modelling of a project is to break it down into a set of subprojects called activities
and then look at how each activity precedes or affects other activities of the project.
The table below shows the activities and their preceding activities in the building of a house and the
duration and number of workers required for each activity.
Activity
Description
Preceding
activity
Duration
(weeks)
Number of
workers
A
Prepare the area and foundation
-
5
3
B
Prepare the pillars and frame
A
8
3
C
Fixing the roof
B
5
2
D
Erecting the walls
B
7
3
E
Exterior plumbing
D
2
4
F
Interior plumbing
E
3
4
G
Electrical work
D
4
3
H
Plastering the walls
F, G
7
4
I
Fixing the ceiling
C
4
4
J
Painting the house
H,I
5
3
K
Flooring
H
4
2
L
Interior fixture
K
4
2
M
Exterior fixture
J
3
2
1. (a) Draw an activity-on-node network for the project.
(b) List down all possible paths and possible durations of the project.
(c) Deduce the critical activities, and state the minimum time for the completion of the project.
2. (a) Construct a table to include the earliest start time (EST), earliest finish time (EFT), latest start
time (LST) and latest finish time (LFT) for each activities.
(b) Determine the total float for each activity.
(c) Identify the critical activities, and determine the minimum time for the completion of the
project.
3. Compare the methods used in 1 and 2.
4. (a) Draw a Gantt chart for the activities, and hence, draw a resource histogram.
(b) Analyse the number of workers required to complete the project in the minimum time.
950/4
50